Problem: If $x+7$ is a factor of $cx^3 + 19x^2 - 3cx + 35$, find the constant $c$.
Answer: Though it is possible to solve this problem using polynomial long division, it is quicker to use the Factor Theorem.

Let $f(x) = cx^3 + 19x^2 - 3cx + 35$. If $x+7$ is a factor of $f(x)$, the factor theorem tells us that $f(-7) = 0.$  Then
\[c(-7)^3 + 19(-7)^2 - 3c(-7) + 35 = 0,\]which simplifies to $-322c + 966 = 0.$  We can solve for $c$ to get $c = \boxed{3}$.